12/03/2019
In meteorology or climate, it is common to pool data or consider problems on a region by region basis.
This can make statistical problems more tractable.
National Resource Management Regions
How should assign regions for the analysis of extremes?
Assign regions that are likely to experience similar impacts
Use these regions to inform our statistical analysis
1. Regionalisation
2. Visualise spatial dependence
3. Spatial post-processing
Form clusters based on extremal dependence!
(Bernard et al 2013)
Use the F-madogram distance (Cooley et al 2006) \[d(x_i, x_j) = \tfrac{1}{2} \mathbb{E} \left[ \left| F_i(M_{x_i}) - F_j(M_{x_j})) \right| \right]\] where \(M_{x_i}\) is the annual maximum rainfall at location \(x_i \in \mathbb{R}^2\) and \(F_i\) is the distribution function of \(M_{x_i}\).
This distance can be estimated non-parametrically.
For \(M_{x_i}\) and \(M_{x_j}\) with common GEV marginals, \(\theta(x_i - x_j)\) is \[\mathbb{P}\left( M_{x_i} \leq z, M_{x_j} \leq z \right) = \left[\mathbb{P}(M_{x_i}\leq z)\mathbb{P}(M_{x_i}\leq z)) \right]^{\tfrac{1}{2}\theta(x_i - x_j)}. %= \exp\left(\dfrac{-\theta(h)}{z}\right),\]
The range of \(\theta(x_i - x_j)\) is \([1 , 2]\).
Can write our distance measure as a function of the extremal coefficient, \(\theta(x_i - x_j)\), \[d(x_i, x_j) = \dfrac{\theta(x_i - x_j) - 1}{2(\theta(x_i - x_j) + 1)}.\]
Therefore the range of \(d(x_i, x_j)\) is \([0 , 1/6]\).
Partitioning around Medoids (PAM): (Kaufman and Rousseeuw 1990)
SWWA EA
Truth
Consider the \(\| x_i - x_j \|\) as the clustering distance.
\[d(C_k, C_{k'}) = \frac{1}{|C_k| |C_{k'}|} \sum_{x_k \in C_k} \sum_{x_{k'} \in C_{k'}} d(x_k, x_{k'}).\]
PICTURE
IMAGE
SHINY APP
IMAGE
Where can we assume a common dependence structure for extremes?
Shiny App
SWWA
TAS
Oesting et. al 2017
approach
cut the region into two